I have known about imaginary numbers for quite some time now. I also understand why we want them to exist (to have a solution for $x^2=-1$). I also remember reading that the complex numbers are closed under addition, multiplication and exponentiation.
What are the quaternions and octonions (I remember seeing $j$ and $k$) and other hypercomplex systems (as they are called), and why did we create them?
Also, I remember reading that the octonions are the largest of these hypercomplex systems (meaning that any number in a hypercomplex system is also a number in the system of the octonions).
Thank you very much in advance.